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Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order. (English) Zbl 0652.35075
Let us consider the mixed problem: \[ (1)\quad u_{tt}- M(\int_{\Omega}| \nabla u| \quad 2dx)\Delta u=f\quad in\quad Q;\quad u(x,0)=u_ 0,\quad u_ t(x,0)=u_ 1\quad in\quad \Omega,\quad u=0\quad on\quad \Sigma. \] By \(\Omega\) we represent a bounded open set of \({\mathbb{R}}^ n,\) \(\Gamma\) its boundary, Q is the cylinder \(\Omega\) \(\times]0,T[\), \(T>0\) and \(\Sigma\) the lateral boundary of Q.
J. L. Lions [cf. Contemporary developments in continuum mechanics and partial differ. equations, Proc. int. Symp. Rio de Janeiro 1977. Math. Stud. 30, 284-346 (1978; Zbl 0404.35002)], motivated by S. I. Pokhozhaev [Mat. Sb., Nov. Ser. 96(138), 152-166 (1975; Zbl 0309.35051)], proposed several questions about (1). One of those questions is to know if there exists some solution of (1) when we choose \(u_ 0\in H\) \(1_ 0(\Omega)\), \(u_ 1\in L\) 2(\(\Omega)\). In the case \(u_ 0\in H\) \(1_ 0(\Omega)\cap H\) 2, \(u_ 1\in H\) \(1_ 0(\Omega)\), there exists a local solution.
The objective of the present work is to give an answer to the above question proposed by J. L. Lions. In fact, we put the system (1) in the form: \[ (2)\quad u''+M(a(u))Au=f,\quad u(0)=u_ 0,\quad u'(0)=u_ 1. \] Here A is the operator defined by \(\{\) ((,));V,H\(\}\), where H,V are Hilbert spaces, \(V\subset H\) continuous and compact. By ((.)) we represent the inner product on V and \(a(u,v)=(Au,v)\), \(a(u)=(Au,u)\). The main result in the present work is to prove that: “If \(u_ 0\in D(A^{3/4})\), \(u_ 1\in D(A^{1/4})\), \(f\in L\) \(2(0,T,D(A^{1/4}))\), then there exists at least one solution for (2).” The authors characterize \(D(A^{3/4})\), \(D(A^{1/4})\) in terms of Sobolev spaces of fractionary order.
Reviewer: L.A.Medeiros

35L70 Second-order nonlinear hyperbolic equations
34G20 Nonlinear differential equations in abstract spaces
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)